[1]} introduced path sampling [2]}
(also called thermodynamic integration) to phylogenetics to address the problem
that the posterior is often dominated by the likelihood and very divergent from
the prior.
Rather than restrict themselves to a sample from the posterior, they collected
MCMC samples from a series of distributions between the prior and posterior.
Specifically, samples are taken from a series of power-posterior distributions,
\( p(D \,|\, T_{}, \theta _{}, M_{})^{\beta }p(T_{}, \theta _{} \,|\, M_{})\) ,
where the likelihood is raised to a power \(\beta \) .
When \(\beta = 1\) , this is equal to the unnormalized joint posterior, which
integrates to what we want to know, the marginal likelihood.
When \(\beta = 0\) , this is equal to the joint prior distribution, which,
assuming we are using proper prior probability distributions, integrates to 1.
If we integrate the power posterior expectation of the derivative
with respect to \(\beta \)  of the log power posterior
over the interval (0–1) with respect to \(\beta \) ,
we get the log ratio of the normalizing constants when \(\beta \) 
equals 1 and 0, and since we know the constant is 1 when \(\beta \)  is zero, we
are left with the marginal likelihood.
[1]} approximated this integral by
summing over MCMC samples taken from a discrete number of \(\beta \)  values evenly
distributed between 1 and 0.
